Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501164 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #23902 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,12)(8,18)(9,22,24,11)(10,14)(13,20,29,36)(15,30)(16,32,23,25)(17,27)(19,33)(21,34)(26,31,28,35);(1,2)(3,4)(6,11)(12,25)(13,18)(14,22)(26,30)(27,32)(28,33)(29,34);(2,4)(5,7,10,17)(6,12,14,27)(8,19,21,15)(9,23,24,16)(11,25,22,32)(13,28,29,26)(18,33,34,30)(20,31,36,35);(5,8,10,21)(6,13,14,29)(7,15,17,19)(9,20,24,36)(11,18,22,34)(12,26,27,28)(16,31,23,35)(25,30,32,33)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,f^b=fh,d^c=di,e^c=ehi,f^d=fh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72312330, #72312352, #72312355, … |
| maximal normal subgroup of | #72312330, #72312352, #72312355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408363);